From Pet to Split

TL;DR

This paper establishes a central limit theorem for sum-product expressions involving mixing processes and polynomial or growth-function indices, extending ergodic theory results to stochastic processes.

Contribution

It introduces the sum-product limit theorem (SPLIT), a new CLT for products of mixing processes evaluated at polynomial or growth functions, generalizing previous ergodic theorems.

Findings

Proves CLT for sums of products of mixing processes with polynomial growth functions.

Applicable to processes from dynamical systems like hyperbolic diffeomorphisms and subshifts.

Extends ergodic theory results to stochastic processes with mixing conditions.

Abstract

Various forms of the polynomial ergodic theorem (PET) which attracted substantial attention in ergodic theory study the limits of expressions having the form $1/N\sum_{n=1}^NT^{q_1(n)}f_1... T^{q_\ell (n)}f_\ell$ where $T$ is a weakly mixing measure preserving transformation, $f_i$'s are bounded measurable functions and $q_i$'s are polynomials taking on integer values on the integers. Motivated partially by these results we obtain a central limit theorem for expressions of the form $1/\sqrt{N}\sum_{n=1}^N (X_1(q_1(n))X_2(q_2(n))... X_\ell(q_\ell(n))-a_1a_2... a_\ell)$ (sum-product limit theorem--SPLIT) where $X_i$'s are fast $\alpha$-mixing bounded stationary processes, $a_j=EX_j(0)$ and $q_i$'s are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when $q_i$'s are polynomials of growing degrees. This result can be applied to the case when $X_i(n)=T^nf_i$ where $T$ is a mixing subshift of finite type, a hyperbolic diffeomorphism or an expanding transformation taken with a Gibbs invariant measure, as well, as to the case when $X_i(n)=f_i(\xi_n)$ where $\xi_n$ is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure.